How to move my files from my old laptop using vsftp

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FTP is a very insecure protocol and has generally been deprecated. There is some configuration necessary to allow non-anonymous access and wider directory access. There should be documentation around. But you would be much better off using a protocol that runs over ssh, such as rsync, sftp, or scp. Why are you wanting to use FileZilla? You don't say what kind of system you are wanting to use to copy the files. If it's Linux, then you can directly connect and copy using the file manager. If it's Windows, then get winscp. I just checked and FileZilla claims to support sftp, so it would work as well.

In Nautilus, you can use CTRL-L to go to a specific file location or any supported url. Or you can click on "Other Locations" at the bottom. Then there will be a text box at the bottom to enter a url with a bit of help text about different available protocols. What you want is sftp://username@hostname_or_ip:/path. Once you're connected, you can make a bookmark for the location to easily access it later by dragging a folder to the left side and dropping it on the "+New Bookmark" label.